Question

1c. use properties of rigid motions to explain why \\( \triangle a b c \cong \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\).

Explanation:

Step1: Define rigid motion

Rigid motion (translation, rotation, reflection) preserves side - lengths and angles.

Step2: Identify the rigid motion

Here, \(\triangle ABC\) is translated (shifted) to get \(\triangle A'B'C'\).
Since translation is a rigid motion, \(AB = A'B'\), \(BC=B'C'\), \(AC = A'C'\) (side - length preservation) and \(\angle A=\angle A'\), \(\angle B=\angle B'\), \(\angle C=\angle C'\) (angle - measure preservation).

Step3: Use the SSS (Side - Side - Side) congruence criterion

In \(\triangle ABC\) and \(\triangle A'B'C'\), \(AB = A'B'\), \(BC = B'C'\), \(AC=A'C'\). By the SSS congruence criterion for triangles (\(SSS\): if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent), \(\triangle ABC\cong\triangle A'B'C'\).

Answer:

\(\triangle ABC\) is translated (a rigid motion) to \(\triangle A'B'C'\). Rigid motions preserve side - lengths and angles. Using the \(SSS\) (Side - Side - Side) congruence criterion (since \(AB = A'B'\), \(BC = B'C'\), \(AC = A'C'\)), \(\triangle ABC\cong\triangle A'B'C'\).